Question: Carlos has taken an initial dose of a prescription medication. The relationship between the elapsed time $t$, in hours, since he took the first dose, and the amount of medication, $M(t)$, in milligrams ( $\text{mg}$ ), in his bloodstream is modeled by the following function. $M(t)=20\cdot e^{{-0.8t}}$ In how many hours will Carlos have $1\text{ mg}$ of medication remaining in his bloodstream? Round your answer, if necessary, to the nearest hundredth.
Thinking about the problem We want to know how many hours it will take, $t$, for the amount of medication in Carlos's bloodstream, $M(t)$, to drop to $1\text{ mg}$. So we need to find the value of $t$ for which $M(t)=1$. Substituting $1$ in for $M(t)$ in the function gives us the following equation. $1=20\cdot e^{{-0.8t}}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}20\cdot e^{-0.8t}&=1\\\\ e^{-0.8t}&=0.05\\\\ -0.8t&=\ln(0.05)\\\\ t&=\dfrac{\ln(0.05)}{-0.8}\\\\ t&\approx 3.74\end{aligned}$ Carlos will have $1\text{ mg}$ of the medication remaining in his blood after $3.74$ hours.